Problem: A circle has a radius of $2$. An arc in this circle has a central angle of $\dfrac{2}{3}\pi$ radians. What is the length of the arc? ${4\pi}$ ${\dfrac{2}{3}\pi}$ $\color{#DF0030}{\dfrac{4}{3}\pi}$ ${2}$
Answer: First, calculate the circumference of the circle. $c = 2\pi r = 2\pi (2) = 4\pi$ The ratio between the arc's central angle $\theta$ and $2 \pi$ radians is equal to the the ratio between the arc length $s$ and the circle's circumference $c$ $\dfrac{\theta}{2 \pi} = \dfrac{s}{c}$ $\dfrac{2}{3}\pi \div 2 \pi = \dfrac{s}{4\pi}$ $\dfrac{1}{3} = \dfrac{s}{4\pi}$ $\dfrac{1}{3} \times 4\pi = s$ $\dfrac{4}{3}\pi = s$